Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $z \neq 0$. $p = \dfrac{9z + 45}{-2z + 8} \div \dfrac{z^2 + 13z + 40}{4z + 32} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{9z + 45}{-2z + 8} \times \dfrac{4z + 32}{z^2 + 13z + 40} $ First factor the quadratic. $p = \dfrac{9z + 45}{-2z + 8} \times \dfrac{4z + 32}{(z + 5)(z + 8)} $ Then factor out any other terms. $p = \dfrac{9(z + 5)}{-2(z - 4)} \times \dfrac{4(z + 8)}{(z + 5)(z + 8)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ 9(z + 5) \times 4(z + 8) } { -2(z - 4) \times (z + 5)(z + 8) } $ $p = \dfrac{ 36(z + 5)(z + 8)}{ -2(z - 4)(z + 5)(z + 8)} $ Notice that $(z + 8)$ and $(z + 5)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ 36\cancel{(z + 5)}(z + 8)}{ -2(z - 4)\cancel{(z + 5)}(z + 8)} $ We are dividing by $z + 5$ , so $z + 5 \neq 0$ Therefore, $z \neq -5$ $p = \dfrac{ 36\cancel{(z + 5)}\cancel{(z + 8)}}{ -2(z - 4)\cancel{(z + 5)}\cancel{(z + 8)}} $ We are dividing by $z + 8$ , so $z + 8 \neq 0$ Therefore, $z \neq -8$ $p = \dfrac{36}{-2(z - 4)} $ $p = \dfrac{-18}{z - 4} ; \space z \neq -5 ; \space z \neq -8 $